We perform a hierarchical basis splitting for the divergence-conforming
space \(\hbox{$V$}_h^r\)  (and \(\hbox{$V$}_h^{r,\mathsf {cst}}\) )
as was done in [1]}, see
also [2]}, to facilitate the discussion on static condensation.
The following basis splitting was presented in [2]}The basis splitting in [2]}
is only performed on triangular meshes,
similar results on other element shapes were documented in [1]}.
These hierarchical bases are readily available in the
NGSolve software[6]}, [7]}, which we use in
our numerical simulations.:
\(\hbox{$V$}_h^r =\bigoplus _{\tiny F\in \mathcal {E}_h} \mathrm {span}\lbrace \hbox{$\phi $}_F^0\rbrace \oplus \!\!\!\!\!\!\!\!\!\!\bigoplus _{\tiny \begin{tabular}{c}F\in \mathcal {E}_h\\i=1,\cdots , n_f^r\end{tabular}}\!\!\!\!\!\!\!\!\!\!\mathrm {span}\lbrace \hbox{$\phi $}_F^i\rbrace \oplus \!\!\!\!\!\!\!\!\!\!\bigoplus _{\tiny \begin{tabular}{c}K\in \mathcal {T}_h\\i=1,\cdots , n_k^{r,1}\end{tabular}}\!\!\!\!\!\!\!\!\!\!\mathrm {span}\lbrace \hbox{$\phi $}_K^{i}\rbrace \oplus \!\!\!\!\!\!\!\!\!\!\bigoplus _{\tiny \begin{tabular}{c}K\in \mathcal {T}_h\\i=1,\cdots , n_k^{r,2}\end{tabular}}\!\!\!\!\!\!\!\!\!\!\mathrm {span}\lbrace \hbox{$\psi $}_K^{i}\rbrace ,\) 
